Frequency locking in Josephson arrays: Connection with the Kuramoto model

The circuit equations for certain series arrays of Josephson junctions can be mapped onto a simple model originally introduced by Kuramoto in Proceedings of the International Symposium on Mathematical Problems in Theoretical Physics, edited by H. Araki, Lecture Notes in Physics Vol. 39 (Springer, Berlin, 1975) to study fundamental aspects of frequency locking in large populations of nonlinear oscillators. This correspondence makes it possible to derive accurate theoretical predictions of transitions signaling the onset of partial and complete locking, respectively. We calculate that both transitions should be observable experimentally using present fabrication tolerances.